Two charges are placed between the plates of a parallel plate capacitor. One charge is +q1 and the other is q2= +5.00x10^-6 C . The charge per unit area on each plate has a magnitude of sigma=1.30 x 10^-4 C/m^2. The force on q1 due to q2 equals the force on q1 due to the electric field of the parallel plate capacitor. What is the distance r between the two charges?

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To find the distance, r, between the two charges, we can equate the force between the charges to the force due to the electric field of the parallel plate capacitor.

First, let's calculate the force between the charges, F_between:

The force between two charges can be calculated using Coulomb's Law:

F_between = k * |q1| * |q2| / r^2

where:
- k is the electrostatic constant, approximately 9 x 10^9 Nm^2/C^2
- q1 and q2 are the magnitudes of the charges
- r is the distance between them

Substituting the given values:
F_between = (9 x 10^9 Nm^2/C^2) * |q1| * |5.00 x 10^-6 C| / r^2

Next, let's calculate the force on q1 due to the electric field of the parallel plate capacitor, F_capacitor:

The force on a charge due to the electric field can be calculated using the formula:

F_capacitor = |q1| * E

where:
- q1 is the magnitude of the charge
- E is the magnitude of the electric field

The electric field between the plates of a parallel plate capacitor is given by:

E = σ / ε₀

where:
- σ is the charge per unit area on each plate
- ε₀ is the permittivity of free space, approximately 8.85 x 10^-12 C^2/Nm^2

Substituting the given values:
E = (1.30 x 10^-4 C/m^2) / (8.85 x 10^-12 C^2/Nm^2)

Now, we can calculate F_capacitor:
F_capacitor = |q1| * (1.30 x 10^-4 C/m^2) / (8.85 x 10^-12 C^2/Nm^2)

Since the problem states that the force on q1 due to q2 equals the force on q1 due to the electric field of the parallel plate capacitor, we can equate F_between to F_capacitor:

(9 x 10^9 Nm^2/C^2) * |q1| * |5.00 x 10^-6 C| / r^2 = |q1| * (1.30 x 10^-4 C/m^2) / (8.85 x 10^-12 C^2/Nm^2)

Simplifying this equation:

(9 x 10^9 Nm^2/C^2) * |q1| * |5.00 x 10^-6 C| / r^2 = |q1| * (1.30 x 10^-4 C/m^2) / (8.85 x 10^-12 C^2/Nm^2)

We can cancel out |q1| and solve for r:

r^2 = [(9 x 10^9 Nm^2/C^2) * (5.00 x 10^-6 C) * (8.85 x 10^-12 C^2/Nm^2)] / [(1.30 x 10^-4 C/m^2)]

Finally, we can take the square root to find r:

r = √[(9 x 10^9 Nm^2/C^2) * (5.00 x 10^-6 C) * (8.85 x 10^-12 C^2/Nm^2)] / [(1.30 x 10^-4 C/m^2)]

Evaluating this expression will give us the value of r.